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Optimal transport (OT) theory focuses, among all maps$T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measureonto another, on those that are the ``thriftiest'', i.e. such that the averagedcost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible.Many computational approaches have been proposed to estimate such Monge mapswhen $c$ is the $\ell_2^2$ distance, e.g., using entropic maps [Pooladian'22],or neural networks [Makkuva'20, Korotin'20]. We propose a new model fortransport maps, built on a family of translation invariant costs $c(x,y):=h(x-y)$, where $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau$ and $\tau$ is aregularizer. We propose a generalization of the entropic map suitable for $h$,and highlight a surprising link tying it with the Bregman centroids of thedivergence $D_h$ generated by $h$, and the proximal operator of $\tau$. We showthat choosing a sparsity-inducing norm for $\tau$ results in maps that applyOccam's razor to transport, in the sense that the displacement vectors$\Delta(x):= T(x)-x$ they induce are sparse, with a sparsity pattern thatvaries depending on $x$. We showcase the ability of our method to estimatemeaningful OT maps for high-dimensional single-cell transcription data, in the$34000$-$d$ space of gene counts for cells, without using dimensionalityreduction, thus retaining the ability to interpret all displacements at thegene level.

Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps